E8 Root System's 240 Vertices and Ising Mass Spectrum Match Toda Theory — E8 Intelligence Research

FINDING: The E8 root system (Gosset 4_21 polytope) encodes 240 vertices and 6720 edges, with 3D projections revealing high-order symmetries; its mass spectrum in the Ising model matches E8 affine Toda field theory predictions. MATH: - E8 root system: 240 roots in 8D, each root vector \( \alpha \) with \( \alpha^2 = 2 \); Cartan matrix determinant = 1 (unimodular lattice). - Gosset 4_21 polytope: 240 vertices, 6720 edges, 17280 faces, 2160 5-faces, 240 7-faces (self-dual). - Coxeter number \( h = 30 \); dual Coxeter number \( h^\vee = 30 \). - Mass ratios from E8 affine Toda (Zamolodchikov): \( m_2/m_1 = 2\cos(\pi/5) = \phi = 1.618... \), \( m_3/m_1 = 2\cos(\pi/30) \cdot \phi \), etc. — the full set: \( m_1, m_2, m_3, m_4, m_5, m_6, m_7, m_8 \) with ratios involving \( \phi \) and \( \sqrt{2+\phi} \). - 3D projection: Coxeter plane projection yields 2-fold, 3-fold, 5-fold rotational symmetries (orders 2, 3, 5) — the icosahedral group \( A_5 \) (order 60) appears in the proje Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com